Combined Model

In statistics world, the combined model may varies in different approach. The combined model that I wanna share in here is combine model for count data. I just want to summarize what the combined model is , but not describe completely since there are a lot of scientific literature out there about it :p (ie: Booth et al( 2003), Molenberghs et al (2007)).

When you have count data in your pocket, you may know that the simplest way to handle such a data with the Poisson distribution model approach. Either you may use complicated or simple model, using the random effect or not.   The Poisson model assumed the equality between mean and variance. Though when we comparing the sample average and sample variance, we might ends up with not get the equality between both statistics. In statistics its called over dispersion/ under dispersion. 

So, whats the solution then? A lot of approaches have been proposed to deal with such a kind of issue. The negative binomial approach was considered to handle over dispersion issue. 

In complicated model *not truly complicated sech :p*, the random effects may included in the model. The simple Poisson model then extended by the so called Poisson-Normal model, if our random effects was assumed follow normal distribution, though we can always change the distribution of random effect *normal dist has been commonly used since its has "DAEBAK" properties :p*.

So when will the combined model takes a part? pls be calm fist pals :p. Now I wanna describe it ;). For Poisson-Normal model, the issue of over/under dispersion still haunted the model. So in order to solve the over/under dispersion issue, we use the negative binomial dist, with normal random effect still remained. Or we could use gamma distribution for random effect. The combination of Poisson-normal model with over/under dispersion gave us the "combined model" . 

ps. Pls give corrections or suggestions to improve this note! thanks a bunch! *_^